The existence of arbitrarily many distinct periodic orbits in a two degree of freedom Hamiltonian system

Melnikov's method is used to prove the existence of arbitrarily many elliptic and hyperbolic periodic orbits in the neighborhood of an elliptic orbit of a two degree of freedom Hamiltonian system which is ‘almost integrable’. The existence of such orbits precludes the existence of analytic second integrals of a certain type. The methods used permit a detailed analysis of the way in which resonant tori break up between the KAM irrational tori which are preserved for weak coupling of two independent nonlinear oscillators.